Let $S$ be a semisimple linear algebraic group $/K$, with $K$ a field and $char K = 0$. Let $H \leq S$ be a closed subgroup $/K$.
Is $H$ semisimple?
2026-05-17 00:23:59.1778977439
Is the closed subgroup of any semisimple linear algebraic group semisimple?
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The answer is no. An algebraic group is semisimple when it has no nontrivial connected normal solvable subgroups. So a nontrivial connected solvable group cannot be semisimple. Certainly $S$ contains some nontrivial connected solvable subgroups..